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ϕΠζਂ૚ֶश ม෼Ϟσϧ ܡɹঘً

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ຊ೔ͷ಺༰ ‣ม෼Ϟσϧ ‣ਖ਼نԽྲྀ ‣֊૚ม෼Ϟσϧ ‣ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏

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ຊ೔ͷ಺༰ ‣ม෼Ϟσϧ ‣ਖ਼نԽྲྀ ‣֊૚ม෼Ϟσϧ ‣ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏

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ม෼Ϟσϧ

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ม෼Ϟσϧ ɹม෼ਪ࿦๏ʹΑΔࣄޙ෼෍ͷۙࣅਪ࿦͸ɼۙࣅ෼෍ ʹରͯ͠ͲͷΑ͏ʹઃܭ͢Δ͔͕ ΞϧΰϦζϜͷੑೳΛࠨӈ͢Δɽ q ۙࣅ෼෍ͷઃܭͰॏཁͳ఺ ᶃɹ Λ࢖ͬͨظ଴஋ܭࢉ΍αϯϓϦϯά͕ߦ͍΍͍͢ ᶄɹ ͕,-μΠόʔδΣϯεͳͲͷࢦඪͷ΋ͱͰ࠷దԽ͠΍͍͢ ᶅɹ ͕ෳࡶͳਅͷࣄޙ෼෍Λਫ਼౓ྑۙ͘ࣅͰ͖ΔΑ͏ͳॊೈ͞Λ΋͍ͬͯΔ q q q

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ม෼Ϟσϧ ۙࣅ෼෍ͷઃܭͰॏཁͳ఺ ᶃɹ Λ࢖ͬͨظ଴஋ܭࢉ΍αϯϓϦϯά͕ߦ͍΍͍͢ ᶄɹ ͕,-μΠόʔδΣϯεͳͲͷࢦඪͷ΋ͱͰ࠷దԽ͠΍͍͢ ᶅɹ ͕ෳࡶͳਅͷࣄޙ෼෍Λਫ਼౓ྑۙ͘ࣅͰ͖ΔΑ͏ͳॊೈ͞Λ΋͍ͬͯΔ q q q ɹฏۉ৔ۙࣅ͸ɼ͜ͷ఺ʹ͍ͭͯͲ͏ͩΖ͏ɾɾɾʁ ᶃ ຬ͍ͨͯ͠Δɽ ʢཧ༝ʣࢦ਺ܕ෼෍ͳͲͷಛੑ͕Α͘஌ΒΕͨ෼෍Λۙࣅͱͯ͠༻͍Δ͔Βɽ

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ม෼Ϟσϧ ۙࣅ෼෍ͷઃܭͰॏཁͳ఺ ᶃɹ Λ࢖ͬͨظ଴஋ܭࢉ΍αϯϓϦϯά͕ߦ͍΍͍͢ ᶄɹ ͕,-μΠόʔδΣϯεͳͲͷࢦඪͷ΋ͱͰ࠷దԽ͠΍͍͢ ᶅɹ ͕ෳࡶͳਅͷࣄޙ෼෍Λਫ਼౓ྑۙ͘ࣅͰ͖ΔΑ͏ͳॊೈ͞Λ΋͍ͬͯΔ q q q ɹฏۉ৔ۙࣅ͸ɼ͜ͷ఺ʹ͍ͭͯͲ͏ͩΖ͏ɾɾɾʁ ᶃ ຬ͍ͨͯ͠Δɽ ʢཧ༝ʣࢦ਺ܕ෼෍ͳͲͷಛੑ͕Α͘஌ΒΕͨ෼෍Λۙࣅͱͯ͠༻͍Δ͔Βɽ ᶄ ຬ͍ͨͯ͠Δɽ ʢཧ༝ʣࣄલ෼෍ͱಉ͡ܗࣜΛۙࣅ෼෍ͱͯ͠બͿ͜ͱͰ&-#0ͷܭࢉΛղੳతʹ͍ͯ͠Δ͔Βɽ

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ม෼Ϟσϧ ۙࣅ෼෍ͷઃܭͰॏཁͳ఺ ᶃɹ Λ࢖ͬͨظ଴஋ܭࢉ΍αϯϓϦϯά͕ߦ͍΍͍͢ ᶄɹ ͕,-μΠόʔδΣϯεͳͲͷࢦඪͷ΋ͱͰ࠷దԽ͠΍͍͢ ᶅɹ ͕ෳࡶͳਅͷࣄޙ෼෍Λਫ਼౓ྑۙ͘ࣅͰ͖ΔΑ͏ͳॊೈ͞Λ΋͍ͬͯΔ q q q ɹฏۉ৔ۙࣅ͸ɼ͜ͷ఺ʹ͍ͭͯͲ͏ͩΖ͏ɾɾɾʁ ᶃ ຬ͍ͨͯ͠Δɽ ʢཧ༝ʣࢦ਺ܕ෼෍ͳͲͷಛੑ͕Α͘஌ΒΕͨ෼෍Λۙࣅͱͯ͠༻͍Δ͔Βɽ ᶄ ຬ͍ͨͯ͠Δɽ ʢཧ༝ʣࣄલ෼෍ͱಉ͡ܗࣜΛۙࣅ෼෍ͱͯ͠બͿ͜ͱͰ&-#0ͷܭࢉΛղੳతʹ͍ͯ͠Δ͔Βɽ ᶅ ຬ͍ͨͯ͠ͳ͍ɽ ʢཧ༝ʣۙࣅ͢Δݸʑͷ֬཰ม਺ʹରͯ͠γϯϓϧͳಠཱੑΛԾఆ͍ͯ͠Δ͔Βɽ

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ม෼Ϟσϧ ۙࣅ෼෍ͷઃܭͰॏཁͳ఺ ᶃɹ Λ࢖ͬͨظ଴஋ܭࢉ΍αϯϓϦϯά͕ߦ͍΍͍͢ ᶄɹ ͕,-μΠόʔδΣϯεͳͲͷࢦඪͷ΋ͱͰ࠷దԽ͠΍͍͢ ᶅɹ ͕ෳࡶͳਅͷࣄޙ෼෍Λਫ਼౓ྑۙ͘ࣅͰ͖ΔΑ͏ͳॊೈ͞Λ΋͍ͬͯΔ q q q ɹฏۉ৔ۙࣅ͸ɼ͜ͷ఺ʹ͍ͭͯͲ͏ͩΖ͏ɾɾɾʁ ᶃ ຬ͍ͨͯ͠Δɽ ʢཧ༝ʣࢦ਺ܕ෼෍ͳͲͷಛੑ͕Α͘஌ΒΕͨ෼෍Λۙࣅͱͯ͠༻͍Δ͔Βɽ ᶄ ຬ͍ͨͯ͠Δɽ ʢཧ༝ʣࣄલ෼෍ͱಉ͡ܗࣜΛۙࣅ෼෍ͱͯ͠બͿ͜ͱͰ&-#0ͷܭࢉΛղੳతʹ͍ͯ͠Δ͔Βɽ ᶅ ຬ͍ͨͯ͠ͳ͍ɽ ʢཧ༝ʣۙࣅ͢Δݸʑͷ֬཰ม਺ʹରͯ͠γϯϓϧͳಠཱੑΛԾఆ͍ͯ͠Δ͔Βɽ ฏۉ৔ۙࣅΛར༻ͨ͠7"&ͷ෼෍ͷۙࣅೳྗ͸ɼ͔ͳΓ੍ݶ͞ΕΔɽ ⟹

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ม෼Ϟσϧ ม෼Ϟσϧ ɹม෼ਪ࿦๏ͷۙࣅ෼෍ ʹ࢖༻͢Δ֬཰෼෍ͷ଒ͷ͜ͱɽ ɹ ੜ੒Ϟσϧɿ؍ଌσʔλͷੜ੒աఔΛදݱ͢Δ֬཰෼෍ɽ q ⇔ ɹฏۉ৔ۙࣅ΍ਪ࿦ωοτϫʔΫʢΤϯίʔμʣ΋ม෼ϞσϧͷҰछͱߟ͑Δ͜ͱ͕Ͱ ͖Δɽ

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ຊ೔ͷ಺༰ ‣ม෼Ϟσϧ ‣ਖ਼نԽྲྀ ‣֊૚ม෼Ϟσϧ ‣ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏

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ਖ਼نԽྲྀ ʲฏۉ৔ۙࣅʹجͮ͘7"&ͷֶशͷ໰୊఺ʳ ɹۙࣅ෼෍ ʹର֯Ψ΢ε෼෍ͳͲͷ୯७ͳ෼෍ΛԾఆ͍ͯ͠Δ͜ͱɽҰൠత ʹෳࡶͳϞσϧʢFHਂ૚ੜ੒Ϟσϧʣͷજࡏม਺ͷਅͷࣄޙ෼෍͸ෳࡶͳ΋ͷʹͳ Δɽ ΑΓෳࡶͳදݱ෼෍Λ΋ͭۙࣅ෼෍Λߟ͑Α͏ʂ q(zn ; xn , ψ) ⟹

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ਖ਼نԽྲྀ ʲฏۉ৔ۙࣅʹجͮ͘7"&ͷֶशͷ໰୊఺ʳ ɹۙࣅ෼෍ ʹର֯Ψ΢ε෼෍ͳͲͷ୯७ͳ෼෍ΛԾఆ͍ͯ͠Δ͜ͱɽҰൠత ʹෳࡶͳϞσϧʢFHਂ૚ੜ੒Ϟσϧʣͷજࡏม਺ͷਅͷࣄޙ෼෍͸ෳࡶͳ΋ͷʹͳ Δɽ ΑΓෳࡶͳදݱ෼෍Λ΋ͭۙࣅ෼෍Λߟ͑Α͏ʂ q(zn ; xn , ψ) ⟹ ਖ਼نԽྲྀʢOPSNBMJ[JOHqPXʣ ɹ؆୯ͳ֬཰෼෍͔Βͷαϯϓϧ ʹରͯ͠ɼෳ਺ճͷՄٯ͔ͭඍ෼Մೳͳؔ਺ ʹΑΔม׵Λద༻͢Δ͜ͱͰɼΑΓෳࡶͳ෼෍͔Βͷαϯϓϧ ΛಘΔ ख๏ɽ w w w w w w w w w w w w w w w z0 f1 , …, fK zK

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ਖ਼نԽྲྀ ʲՄٯͳؔ਺ʹΑΔม׵ʳ ɹՄٯͰ࿈ଓͳؔ਺ Λߟ͑Δɽม׵ Λ༻͍Ε͹ɼ֬཰ີ౓ؔ਺ ʹରͯ͠ ͸ҎԼͷΑ͏ʹͳΔɽʢ֬཰ີ౓ؔ਺ͷม׵͸અࢀরʣ ͓Αͼ ͸ϠίϏߦྻɽ ͸ߦྻࣜɽ ͜ͷม׵Λ ͔Β ճద༻͢Δ͜ͱΛߟ͑Δɽ ͕ͨͬͯ͠ɼ࠷ऴతͳ֬཰ม਺ ͷີ౓ؔ਺͸ҎԼͷΑ͏ʹͳΔɽ ɹɹɹɹɹɹɹɹɹɹɹ f : ℝD → ℝD ̂ z = f(z) q(z) q( ̂ z) q( ̂ z) = q(z) det ( ∂f−1 ∂ ̂ z ) = q(z) det ( ∂f ∂z) −1 ∂f−1 ∂ ̂ z ∂f ∂z det( ⋅ ) z0 K zK = fK ∘ ⋯ ∘ f1 (z0 ) zK qK (zK ) = q0 (z0 ) K ∏ k=1 det ( ∂fk ∂zk−1 ) −1

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ਖ਼نԽྲྀ ʲม׵ͷྫʳ ɹฏ໘ྲྀʢQMBOBSqPXʣɹ͸ؔ਺Λ࣍ͷΑ͏ʹ͢Δɽ ɹ ͸ඍ෼Մೳͳඇઢܗؔ਺ɽ ͸ม׵ΛܾΊΔύϥϝʔλɽ ม෼ਪ࿦๏Ͱ͸ɼ ͸ม෼ύϥϝʔλͷ໾ׂΛՌͨ͢ɽฏ໘ྲྀʹΑͬͯಘΒΕ෼෍ͷີ ౓ܭࢉʹඞཁͳϠίϏߦྻ͸ ͰܭࢉͰ͖Δɽ ͨͩ͠ɼ ͷಋؔ਺Λ ͱ͓͍ͨɽ ɹ ʹରͯؔ͠਺Λ܁Γฦ͠ద༻ͯ͠ಘΒΕΔີ౓ؔ਺͸ɼ௒ฏ໘ ʹਨ ௚ͳํ޲ʹऩॖͱ֦େΛ܁Γฦ͍͖ͯ͠ɼ࠷ऴతʹಘΒΕΔ ͸ෳࡶͳ෼෍Λܗ੒͢ Δɽ f f(z) = z + uh(wTz + b) . h λ = {w ∈ ℝD, u ∈ ℝD, b ∈ ℝ} λ (D) det ( ∂f ∂z) = |1 + uTψ(z)| h ψ(z) = h′(wTz + b)w z0 f wTz + b = 0 zK

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ਖ਼نԽྲྀ ʲม׵ͷྫʳ ɹ์ࣹঢ়ྲྀʢSBEJBMqPXʣɹ͸ج४ͱͳΔ఺ͷपลͰҎԼͷΑ͏ͳؔ਺Λ༻͍ͯ ֬཰ີ౓Λม׵͢Δɽ ɹ ͨͩ͠ɼ ɼ ͱ͢Δɽύϥϝʔλ͸ ์ࣹঢ়ྲྀͷϠίϏߦྻ͸ҎԼͷΑ͏ʹ؆୯ʹܭࢉͰ͖Δɽ ̂ z f f(z) = z + βh(α, r)(z − ̂ z) . r = |z − ̂ z| h(α, r) = 1 α + r λ = { ̂ z ∈ ℝD, α ∈ ℝ, β ∈ ℝ} det ( ∂f ∂z ) = {1 + βh(α, r)}D−1{1 + βh(α, r) + βh′(α, r)r} ʢ͜ͷϠίϏߦྻͬͯɼͲ͏ٻΊͯΔʁʣ

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ਖ਼نԽྲྀ ʲม׵ͷྫʳ ग़యɿ“Variational Inference with Normalizing Flows”, Danilo J. Rezende and Shakir Mohamed., ICML, 2015

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ਖ਼نԽྲྀ ʲม෼ਪ࿦๏΁ͷద༻ʳ ฏۉ৔ۙࣅʹجͮ͘ม෼ਪ࿦๏ ୯७ͳԾఆʹΑΓෳࡶͳ෼෍ͷۙࣅੑೳ͕ѱ͍ɽ ⟹

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ਖ਼نԽྲྀ ʲม෼ਪ࿦๏΁ͷద༻ʳ ฏۉ৔ۙࣅʹجͮ͘ม෼ਪ࿦๏ ୯७ͳԾఆʹΑΓෳࡶͳ෼෍ͷۙࣅੑೳ͕ѱ͍ɽ ม෼ਪ࿦๏ʹਖ਼نԽྲྀΛ૊Έ߹ΘͤΔ ฏۉ৔ۙࣅʹΑΔਪ࿦ΑΓ΋͸Δ͔ʹਫ਼౓ͷߴ͍ࣄޙ෼෍ͷۙࣅ͕Մೳɽ ⟹ ⟹

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ਖ਼نԽྲྀ ʲม෼ਪ࿦๏΁ͷద༻ʳ ɹજࡏม਺ͷू߹Λ ͱͨ͠ͱ͖ɼੜ੒Ϟσϧ ʹରͯ͠ɼ ਖ਼نԽྲྀΛద༻ͨ͠৔߹ɼ͋Δσʔλ ʹର͢Δม෼ΤωϧΪʔ͸ɼҎԼͷΑ͏ʹͳ Δɽ Z p(X, Z) = N ∏ n=1 p(xn |zn )p(zn ) x ℱ[q] = q(z) [ln q(z) − ln p(x, z)] = q0 (z0 ) [ln qK (zK ) − ln p(x, zK )] = q0 (z0 ) [ln q0 (z0 )] − Eq0 (z0 ) [ln p(x, zK )] − Eq0 (z0 ) K ∑ k=1 ln det ( ∂fk ∂zk−1 )

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ਖ਼نԽྲྀ ʲม෼ਪ࿦๏΁ͷద༻ʳ ɹ7"&ʹద༻͢Δ৔߹͸ɼॳظ෼෍ΛҎԼͷΑ͏ʹ͢Δɽ ਖ਼نԽྲྀͷύϥϝʔλ ΋//ͷग़ྗΛ༻͍Δ͜ͱ͕Ͱ͖ΔɽΤϯίʔμΛ࢖ͬͯޮ཰ తʹજࡏม਺શମͷۙࣅ෼෍Λֶश͠ɼਖ਼نԽྲྀͰෳࡶͳ෼෍ʹม׵͢Δ͜ͱͰɼਫ਼౓ ͷߴ͍ۙࣅΛߦ͏͜ͱ͕ՄೳͱͳΔɽ q0 (z0 ) = (z|m(x; ψ), diagm(v(x; ψ))) λ ग़యɿ“Variational Inference with Normalizing Flows”, Danilo J. Rezende and Shakir Mohamed., ICML, 2015 normalizing flow

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ਖ਼نԽྲྀ ʲޡهʳࣜʢʣʹؔΘΔ෦෼ ɹຊʹ͸ɼࣜʢʣ͸&-#0ͷܭࢉʹ ͳ͍ͬͯΔ͕ɼ͜Ε͸ม෼ΤωϧΪʔɽ ℒ[q] → ℱ[q] = − ℒ[q] ʢݩ࿦จ͔Βൈਮʣ

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ਖ਼نԽྲྀ ʲม෼ਪ࿦๏΁ͷద༻ʳม෼ΤωϧΪʔͷಋग़ qK (zK ) = q0 (z0 ) K ∏ k=1 det ( ∂fk ∂zk ) −1 ln qK (zK ) = ln q0 (z0 ) + K ∑ k=1 ln det ( ∂fk ∂zk ) −1 = ln q0 (z0 ) − K ∑ k=1 ln det ( ∂fk ∂zk ) ℱ[q] = − ℒ[q] = − ∫ q(z)ln p(x, z) q(z) dz = ∫ q(z)ln q(z) p(x, z) dz = q(z) [ln q(z) − ln p(x, z)] = ∫ q0 (z0 )ln qK (zK ) p(x, zK ) dz = q0 (z0 ) [ln qK (zK ) − ln p(x, zK )] ( ∵ normalizing flow) = q0 (z0 ) ln q0 (z0 ) − K ∑ k=1 ln det ( ∂fk ∂zk ) − ln p(x, zK ) = q0 (z0 ) [ln q0 (z0 )] − q0 (z0 ) [ln p(x, zK )] − q0 (z0 ) K ∑ k=1 ln det ( ∂fk ∂zk−1 )

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ਖ਼نԽྲྀ ʲελΠϯม෼ޯ഑߱Լ๏ʳ ɹ ελΠϯม෼ޯ഑߱Լ๏ ɹஞ࣍తͳม਺ม׵Λར༻ͨ͠ม෼ਪ࿦๏ɽ࠶ੜ֩ώϧϕϧτ্ۭؒͰͷ൚ؔ਺ ඍ෼Λར༻ͨ͠ޯ഑߱Լ๏Λద༻͢Δ͜ͱͰɼਅͷࣄޙ෼෍ʹର͢Δ,-μΠόʔ δΣϯεΛ࠷খԽ͢Δख๏ɽ ɹۙࣅࣄޙ෼෍͸ɼॳظ෼෍͔Βͷ༗ݶݸͷαϯϓϧ͔Βදݱ͞Εɼ࠷దԽʹΑͬͯ ਅͷࣄޙ෼෍͔Βͷαϯϓϧʹม׵͞ΕΔɽ ʲར఺ʳߦྻࣜ΍ٯߦྻͷܭࢉ͕ෆཁͳ఺ɽ ɹɹɹɹɹɹɹɹɹɹɹɹɹ ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹ࠶ੜ֩ώϧϕϧτۭؒʹର͢Δ஌͕ࣝͳ͘ʜ

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ຊ೔ͷ಺༰ ‣ม෼Ϟσϧ ‣ਖ਼نԽྲྀ ‣֊૚ม෼Ϟσϧ ‣ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏

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֊૚ม෼Ϟσϧ ʲۙࣅ෼෍ͷϞσϧʳ ɹฏۉ৔ۙࣅΛ༻͍ͨજࡏม਺ ͷۙࣅ෼෍Λ ͱ͢Δɽɹ ɹɹɹɹɹɹɹɹɹɹɹɹɹɹ ɿม෼ύϥϝʔλͷू߹ ฏۉ৔ۙࣅͰ͸ɼ.ݸͷજࡏม਺͸ಠཱ͍ͯ͠ΔͱԾఆ͍ͯ͠Δɽ ɹɹɹɹɹɹɹɹɹɹɹҰํɼ֊૚ม෼Ϟσϧ͸ɾɾɾʁ Z = {z1 , …, zM } qMF qMF (Z; λ) = M ∏ m=1 q(zm ; λm ) λ ֊૚ม෼Ϟσϧ ɹิॿજࡏม਺๏ͱ΋ݺ͹Ε͍ͯΔɽม෼ϞσϧͷҰछͰɼۙࣅ෼෍Λ֊૚Խ͢ Δ͜ͱʹΑΓෳࡶͳۙࣅ෼෍ΛදݱͰ͖ΔΑ͏ʹ֦ுͨ͠΋ͷɽ w w w

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֊૚ม෼Ϟσϧ ʲۙࣅ෼෍ͷϞσϧʳ ɹ֊૚ม෼ϞσϧʹΑΔۙࣅ෼෍ ͸࣍ͷΑ͏ͳܗࣜΛͱΔ͜ͱͰɼۙࣅ෼෍Λ֊ ૚Խ͢Δɽ Λม෼ࣄલ෼෍ɼ Λม෼໬౓ͱݺͿɽ ʹؔͯ͠पลԽ͢Δ͜ͱͰɼۙࣅ ෼෍͸͋Δछͷࠞ߹෼෍ʹͳΔɽ ɹม෼ύϥϝʔλͷੜ੒ʹಠཱͰͳ͍෼෍ΛԾఆ͢Δ͜ͱͰɼજࡏม਺ؒͷ૬ؔΛଊ͑ Δ͜ͱ͕Ͱ͖ΔΑ͏ʹͳΔɽ֊૚ม෼Ϟσϧ͸ɼม෼ύϥϝʔλʹؔͯ͠&-#0࠷େ Խ͢Δ͜ͱͰɼม෼ਪ࿦๏ͷ࿮૊ΈͰֶश͕Մೳɽ qHVM qHVM (Z; ξ) = ∫ q(λ; ξ) M ∏ m=1 q(zm ; λm )dλ q(λ; ξ) q(zm |λm ) λ ξ ֊૚ม෼Ϟσϧ ɹิॿજࡏม਺๏ͱ΋ݺ͹Ε͍ͯΔɽม෼ϞσϧͷҰछͰɼۙࣅ෼෍Λ֊૚Խ͢ Δ͜ͱʹΑΓෳࡶͳۙࣅ෼෍ΛදݱͰ͖ΔΑ͏ʹ֦ுͨ͠΋ͷɽ https://arxiv.org/abs/1511.02386

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֊૚ม෼Ϟσϧ ʲۙࣅ෼෍ͷϞσϧʳ ֊૚ม෼Ϟσϧ ɹิॿજࡏม਺๏ͱ΋ݺ͹Ε͍ͯΔɽม෼ϞσϧͷҰछͰɼۙࣅ෼෍Λ֊૚Խ͢ Δ͜ͱʹΑΓෳࡶͳۙࣅ෼෍ΛදݱͰ͖ΔΑ͏ʹ֦ுͨ͠΋ͷɽ ฏۉ৔ۙࣅ ֊૚ม෼Ϟσϧ m = 1,…, M zm λm m = 1,…, M zm λm ξ

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֊૚ม෼Ϟσϧ ʲม෼ࣄલ෼෍ͷྫʳ ɹ Λࠞ߹ཁૉ਺ɼ Λ ࣍ݩͷΧςΰϦ෼෍ͷύϥϝʔλɼ Λ ࣍ݩΨ ΢ε෼෍ͷύϥϝʔλͷू߹ͱ͢Ε͹ɼҎԼͷΑ͏ʹࠞ߹Ϟσϧͷม෼ࣄલ෼෍Λߟ͑ ΒΕΔɽ ݁Ռɼજࡏม਺ؒͷৄࡉͳ૬ؔΛଊ͑Δ͜ͱ͕ՄೳͱͳΔɽ ɹ·ͨɼม෼ࣄલ෼෍ʹਖ਼نԽྲྀΛద༻͢Δ͜ͱ΋Մೳɽ K π K ξ = {μk , Σk }K k=1 M q(λ; ξ) = K ∑ k=1 πk (λ|μk , Σk ) q(λ; ξ) = q(λ0 ) K ∏ k=1 det ( ∂f ∂λk−1 ) −1 ֊૚ม෼ϞσϧʹΑΔۙࣅ෼෍ ɹɹɹɹɹɹɹɹɹɹ qHVM = ∫ q(λ; ξ) M ∏ m=1 q(zm ; λm )dλ ม෼ࣄલ෼෍ ม෼໬౓ q(λ; ξ) q(zm ; λm )

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֊૚ม෼Ϟσϧ ʲม෼ࣄલ෼෍ͷྫʳ ɹม෼Ϟσϧͱͯ͠Ψ΢εաఔΛ༻͍Δ͜ͱ΋Մೳɽ ม෼Ψ΢εաఔ ɹ ͸ม෼σʔλͱ͍͏ม෼Ψ΢εաఔͷͨΊͷٖࣅతͳೖग़ྗσʔλɽม෼σʔλͱ ڞ෼ࢄؔ਺ͷύϥϝʔλ ͕ม෼Ψ΢εաఔʹ͓͚Δม෼ύϥϝʔλͰɼ&-#0ʹجͮ ͖࠷దԽɽΨ΢εաఔʹै͏ؔ਺ ʹΑͬͯɼજࡏೖྗ͕ ʹϚοϐϯά͞Εɼม ෼໬౓ ʹΑͬͯਪ࿦͍ͨ͠જࡏม਺ ͷ෼෍͕ܾ·Δɽ ɹ͜ΕΒͷม෼Ϟσϧ͸ɼϒϥοΫϘοΫεม෼ਪ࿦๏ͱݺ͹ΕΔख๏ͳͲͰ&-#0࠷ େԽʹ࢖͑ΔɽϒϥοΫϘοΫεม෼ਪ࿦๏͸είΞؔ਺ਪఆʹجͮ͘&-#0ͷޯ഑ۙ ࣅख๏ɽ ⟹ qVGP (Z; θ, V) = ∫ ∫ M ∏ m=1 q(zm |Fm (ξ))(Fm ; O, Kξ,ξ )(ξ; 0, I)dFdξ V θ Fm ξ Fm (ξ) q(zm |Fm (ξ)) zm https://arxiv.org/abs/1511.06499 ষΛಡΜͩޙʹಡΈฦ͢ͷ͕ྑͦ͞͏ɽ

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ຊ೔ͷ಺༰ ‣ม෼Ϟσϧ ‣ਖ਼نԽྲྀ ‣֊૚ม෼Ϟσϧ ‣ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ɹ͍··Ͱ΍͖ͬͯͨ֬཰ੜ੒Ϟσϧ͸ɼີ౓ܭࢉ͕ܭࢉՄೳͳ֬཰෼෍Λ૊Έ߹Θͤ ͯϞσϦϯά͖ͯͨ͠ɽ ɹີ౓ܭࢉͰ͖ͳ͍৔߹ʹ͓͚Δσʔλͷੜ੒ͷݚڀ΋ߦΘΕ͍ͯΔɽ ඇ໌ࣔతϞσϧ ɹີ౓Λܭࢉ͢Δ͜ͱ͕Ͱ͖ͳ͍΋ͷͷɼσʔλͷੜ੒͸ߦ͏͜ͱ͕Ͱ͖ΔΑ͏ ͳϞσϧɽ ɹඇ໌ࣔతϞσϧͷऔΓѻ͍͸ɼۙࣅϕΠζܭࢉͱͯ͠௕͘ݚڀ͞Ε͍ͯΔɽ ໬౓ͳ͠ม෼ਪ࿦๏ ɹੜ੒Ϟσϧ΍ۙࣅ෼෍͕ඇ໌ࣔతϞσϧͱͯ͠ߏ੒͞Ε͍ͯΔঢ়گΛ૝ఆͨ͠ਪ ࿦ΞϧΰϦζϜɽ ɹ໬౓ͳ͠ม෼ਪ࿦๏ʹΑΔੜ੒Ϟσϧͷֶशํ๏͸ɼఢରతੜ੒ωοτϫʔΫ ʢ("/ʣʹ΋ར༻͞Ε͍ͯΔɽ ɹ ɹҎ߱ɼʮඇ໌ࣔతʯ͸ʮີ౓ܭࢉ͕Ͱ͖ͳ͍ʯ͜ͱΛࢦ͢ɽ

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲඇ໌ࣔతϞσϧʳ ɹ࣍ͷΑ͏ͳ؍ଌσʔλ ͷ֊૚తͳੜ੒ϞσϧΛߟ͑Δɽ ɹɹɹɹɹɹɹɹɹɹɹɹ ɿજࡏม਺ ɹɹɹɹɹɹɹɹɹɹɹɹ ɿશͯͷσʔλͰڞ༗͞ΕΔύϥϝʔλͷू߹ ɹ ͸ඇ໌ࣔతͳ෼෍Ͱ͋Δͱఆٛ͢Δɽͭ·Γɼ ɹ্ͷࣜͷΑ͏ʹؔ਺ ͱϊΠζ ʹΑͬͯσʔλ ͕ੜ੒͞ΕΔͱ͢Ε͹ɼҎԼͷ Α͏ʹ໬౓͕ܭࢉͰ͖Δɽ ɹ͜ͷੵ෼͸ղੳతʹܭࢉෆՄͰɼޮ཰తʹ໬౓ܭࢉ͕Ͱ͖ͳ͍ͱԾఆɽ·ͨɼύϥ ϝʔλͷࣄલ෼෍ ͸αϯϓϦϯά΋ີ౓ܭࢉ΋༰қͱ͢Δɽ X p(X, Z, θ) = p(θ) N ∏ n=1 p(xn |zn , θ)p(zn |θ) Z θ p(xn |zn , θ) ϵn ∼ p(ϵ) xn = g(ϵn |zn , θ) g ϵn xn p(xn ∈ A|zn , θ) = ∫ xn ∈A p(ϵn )dϵn p(θ) https://arxiv.org/abs/1702.08896

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲඇ໌ࣔతϞσϧʳ ֊૚Ϟσϧ n = 1,…, N θ zn xn ඇ໌ࣔత֊૚Ϟσϧ n = 1,…, N θ zn xn ϵn ਖ਼ํܗ͸ɼܾఆతؔ਺

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ ɹඇ໌ࣔతϞσϧͷࣄޙ෼෍͸ҎԼͷΑ͏ʹͳΔɽ ͔͠͠ɼ͜Ε͸ղੳతʹܭࢉͰ͖ͳ͍ɽ ɹ ม෼ਪ࿦๏ʹΑΔࣄޙ෼෍ͷۙࣅɽ ɹҰൠతʹඇ໌ࣔతϞσϧ͸ࣄޙ෼෍΋ෳࡶʹͳΔͷͰɼԾఆ͢Δۙࣅ෼෍΋දݱྗ͕ ߴ͍΄͏͕ྑ͍ɽ ɹ ۙࣅ෼෍ʹԾఆ͢Δ੍໿ΛऑΊɼΑΓ޿͍Ϋϥεͷۙࣅ෼෍Λઃఆɽ p(Z, θ|X) = p(X, Z, θ) p(X) ⟹ ⟹

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ ɹજࡏม਺ͷۙࣅ෼෍ʹରͯ͠΋ม෼ύϥϝʔλΛ ͱͨ͠ඇ໌ࣔతͳ෼෍ΛԾఆɽ જࡏม਺ ͸؆୯ʹαϯϓϧՄೳɽม෼໬౓ ͷ஋ࣗମ͸ܭࢉͰ͖ͳͯ͘΋ྑ ͍ͱ͢Δɽ w w w w w w w ψ zn ∼ qψ (zn |xn , θ) zn qψ (zn |xn , θ) ໬౓ͳ͠ม෼ਪ࿦๏ͷ໨త ɹ໌ࣔతͳີ౓ؔ਺Λ࣋ͨͣɼαϯϓϧ ΛಘΒΕΔ͜ͱ͚ͩ Λར༻ͯ͠ม෼ਪ࿦๏Λ࣮ߦ͢Δ͜ͱɽ zn n = 1,…, N θ zn xn ϵn ψ

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ ɹඇ໌ࣔతͳม෼໬౓ͱɼม෼ύϥϝʔλΛ΋ͭ ͷۙࣅ෼෍ Λ༻͍ͯۙࣅࣄޙ ෼෍શମΛҎԼͷΑ͏ʹ͢Δɽ ͸ɼ ͷαϯϓϦϯά΋ີ౓ܭࢉ΋༰қͳ֬཰ີ౓ؔ਺ʢFHΨ΢ε෼෍ʣΛઃఆ ͢Δɽ ξ θ qξ (θ) qψ,ξ (Z, θ|X) = qξ (θ) N ∏ n=1 qψ (zn |xn , θ) qξ (θ) θ n = 1,…, N θ zn xn ϵn ψ ξ

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ ɹ্ͷۙࣅࣄޙ෼෍ΑΓɼର਺पล໬౓ͷ&-#0͸ҎԼͷΑ͏ʹॻ͚Δɽ ͓Αͼ ͸ඇ໌ࣔతͳ෼෍ɽ ɹ ޯ഑߱Լ๏ͳͲͰ&-#0࠷େԽ͕Ͱ͖ͳ͍ɽ ʲղܾࡦʳ ɹσʔλͷܦݧ෼෍ Λར༻͢Δɽ qψ,ξ (Z, θ|X) = qξ (θ) N ∏ n=1 qψ (zn |xn , θ) ℒ(ψ, ξ) = qψ,ξ (Z,θ|X) [ln p(X, Z, θ) − ln qψ,ξ (Z, θ|X)] = qξ (θ) [ln p(θ) − ln qξ (θ)] + N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [ln p(xn , zn |θ) − ln qψ (zn |xn , θ)] p(xn , zn |θ) qψ (zn |xn , θ) ⟹ q (xn )

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ ɹͲͷΑ͏ʹ Λར༻͢Δ͔ʁ ʹ ΛՃ͑Δɽ ɹ໬౓ͳ͠ม෼ਪ࿦๏Ͱ͸ɼඇ໌ࣔతͳ෼෍Λ௚઀ѻ͏୅ΘΓʹɼີ౓ൺͷର਺Λ௚઀ ਪఆ͢Δ͜ͱͰԼքͷܭࢉΛߦ͏ɽ ɹີ౓ਪఆث ʹ͸ɼύϥϝʔλΛ ͱͨ͠ඍ෼Մೳͳχϡʔϥϧωοτ ϫʔΫͳͲͷճؼϞσϧΛબ୒͞ΕΔɽ q (xn ) ⟹ ℒ(ψ, ξ) −ln q (xn ) ℒ(ψ, ξ) = qξ (θ) [ln p(θ) − ln qξ (θ)] + N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [ ln p(xn , zn |θ) qψ, (xn , zn |θ) ] + c ropt. (xn , zn , θ|η) = ln p(xn , zn |θ) qψ, (xn , zn |θ) r(xn , zn , θ|η) η

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ ɹີ౓ਪఆثͷֶशͷྫͱͯ͠ɼదਖ਼είΞنଇʹج͍ͮͨଛࣦؔ਺Λ࢖༻͢Δ͜ͱ ͕ڍ͛ΒΕΔɽ ɹɹɹɹɹɹɹɹɹ ͕ ͔Βͷαϯϓϧʹରͯ͠ɼ ͔Βͷαϯϓϧʹରͯ͠Λฦ͢ͱ ͖ɼ ΛͱΔɽ ɹͭ·Γɼີ౓ਪఆث ͸ ͓Αͼ ͔ΒͷαϯϓϧͷΈ࢖ͬͯɼ ͷ ͷ ޯ഑ʹؔ͢ΔෆภਪఆྔΛಘΔ͜ͱʹΑΓֶशͰ͖Δɽ r J(η) = p(xn ,zn |θ) [−ln Sig(r(xn , zn , θ|η))] +qψ (xn ,zn |θ) [−ln{1 − Sig(r(xn , zn , θ|η))}] Sig(r(xn , zn , θ|η)) p q J(η) = 0 r(xn , zn , θ|η) xn zn J(η) η

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ ɹΑͬͯɼ࠷େԽ͢Δ໨తؔ਺͸ҎԼͷΑ͏ʹͳΔɽ ɹ ɹ࠶ύϥϝʔλԽޯ഑Λ࢖ͬͯɼ ͓Αͼ ΛαϯϓϦϯάͯ͠ɼม෼ύϥϝʔλ ͓ Αͼʹؔ͢Δޯ഑ͷۙࣅΛಘΔɽ ℒr (ψ, ξ) = qξ (θ) [ln p(θ) − ln qξ (θ)] + N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [r(xn , zn , θ|η)] zn θ ψ ξ

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ ໬౓ͳ͠ม෼ਪ࿦๏ͷ·ͱΊ wೖྗɿඇ໌ࣔతϞσϧ ɼࣄલ෼෍ ɼඇ໌ࣔతม෼໬౓ؔ਺ ɼม෼ࣄલ෼෍ ɼີ౓ൺਪఆث wग़ྗɿม෼ύϥϝʔλɹ ɼ wύϥϝʔλ ɼɼ ͷॳظԽ wԼهΛऩଋ͢Δ·Ͱ܁Γฦ͢ɽ ޯ഑ ɼ ɼ ͷෆภਪఆྔΛܭࢉ ɼ ɼΛߋ৽͢Δɽ p(zn , θ|xn ) p(θ) qψ (zn |xn , θ) qξ (θ) r(xn , zn , θ|η) ψ ξ ψ ξ η ∇J(η) ∇ψ ℒ(ψ, ξ) ∇ξ ℒ(ψ, ξ) η ψ ξ

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ໨తؔ਺ͷಋग़ ln p(X, Z, θ) = ln p(X, Z|θ)p(θ) = ln p(X, Z|θ) + ln p(θ) = N ∑ n=1 ln p(xn , zn |θ) + ln p(θ), ln qψ,ξ (Z, θ|X) = ln qξ (θ) + N ∑ n=1 ln qψ (zn |xn , θ), ℒ(ψ, ξ) = ∫ ∫ qψ,ξ (Z, θ|X)ln p(X, Z, θ) qψ,ξ (Z, θ|X) dZdθ = qψ,ξ (Z,θ|X) [ln p(X, Z, θ) − ln qψ,ξ (Z, θ|X)] = qξ (θ)qψ (Z|X,θ) [ln p(X, Z|θ) + ln p(θ) −ln qξ (θ) − ln qψ (Z|X, θ)] = qξ (θ)qψ (Z|X,θ) [ln p(θ) − ln qξ (θ)] +qξ (θ)qψ (Z|X,θ) [ln p(X, Z|θ) − ln qψ (Z|X, θ)] = qξ (θ) [ln p(θ) − ln qξ (θ)] + N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [ln p(xn , zn |θ) − ln qψ (zn |xn , θ)] ͨͩ͠ɼ ℒ(ψ, ξ) = qξ (θ) [ln p(θ) − ln qξ (θ)] + N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [ln p(xn , zn |θ) − ln qψ (zn |xn , θ) −ln p (xn ) + ln p (xn )] = qξ (θ) [ln p(θ) − ln qξ (θ)] + N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [ ln p(xn , zn |θ) qψ (zn |xn , θ)p (xn )] + c = qξ (θ) [ln p(θ) − ln qξ (θ)] + N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [ ln p(xn , zn |θ) qψ, (xn , zn |θ)] + c c = N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [ln p (xn )] = N ∑ n=1 ln p (xn )

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ඇ໌ࣔతϞσϧͱ໬౓ͳ͠ม෼ਪ࿦๏ ʲ໬౓ͳ͠ม෼ਪ࿦๏ʳ໨తؔ਺ͷಋग़ ͱͨ͠ͱ͖ɼ ͱ͓͘ͱɼ ɹɹɹɹ ɹɹɹɹ ͱͳΔͷͰɼ ࠷େԽͱ ࠷େԽ͸౳Ձɽ ͕ͨͬͯ͠ɼ࠷େԽ͍ͨ͠໨తؔ਺͕ ͱͳΔɽ r(xn , zn , θ; η) = ln p(xn , zn |θ) qψ, (xn , zn |θ) ℒr (ψ, ξ) = qξ (θ) [ln p(θ) − ln qξ (θ)] + N ∑ n=1 qξ (θ)qψ (zn |xn ,θ) [r(xn , zn , θ; η)] ∇ψ ℒ(ψ, ξ) = ∇ψ ℒr (ψ, ξ) ∇ξ ℒ(ψ, ξ) = ∇ξ ℒr (ψ, ξ) ℒ(ψ, ξ) ℒr (ψ, ξ) ℒr (ψ, ξ)